In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation:
x y ″ + ( 1 − x ) y ′ + n y = 0 , y = L ( x ) {\displaystyle xy''+(1-x)y'+ny=0,\ y=L(x)}
which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. Sometimes the name Laguerre polynomials is used for solutions of
x y ″ + ( α + 1 − x ) y ′ + n y = 0 . {\displaystyle xy''+(\alpha +1-x)y'+ny=0~.}
where n is still a non-negative integer. Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin). More generally, a Laguerre function is a solution when n is not necessarily a non-negative integer. The Laguerre polynomials are also used for Gauss–Laguerre quadrature to numerically compute integrals of the form
∫ 0 ∞ f ( x ) e − x d x . {\displaystyle \int _{0}^{\infty }f(x)e^{-x}\,dx.}
These polynomials, usually denoted L0, L1, ..., are a polynomial sequence which may be defined by the Rodrigues formula,
L n ( x ) = e x n ! d n d x n ( e − x x n ) = 1 n ! ( d d x − 1 ) n x n , {\displaystyle L_{n}(x)={\frac {e^{x}}{n!}}{\frac {d^{n}}{dx^{n}}}\left(e^{-x}x^{n}\right)={\frac {1}{n!}}\left({\frac {d}{dx}}-1\right)^{n}x^{n},}
reducing to the closed form of a following section. They are orthogonal polynomials with respect to an inner product
⟨ f , g ⟩ = ∫ 0 ∞ f ( x ) g ( x ) e − x d x . {\displaystyle \langle f,g\rangle =\int _{0}^{\infty }f(x)g(x)e^{-x}\,dx.}
The rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables. Further see the Tricomi–Carlitz polynomials. The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom. They also describe the static Wigner functions of oscillator systems in quantum mechanics in phase space. They further enter in the quantum mechanics of the Morse potential and of the 3D isotropic harmonic oscillator. Physicists sometimes use a definition for the Laguerre polynomials that is larger by a factor of n! than the definition used here. (Likewise, some physicists may use somewhat different definitions of the so-called associated Laguerre polynomials.)
Recursive definition, closed form, and generating function One can also define the Laguerre polynomials recursively, defining the first two polynomials as
L 0 ( x ) = 1 {\displaystyle L_{0}(x)=1}
L 1 ( x ) = 1 − x {\displaystyle L_{1}(x)=1-x}
and then using the following recurrence relation for any k ≥ 1:
L k + 1 ( x ) = ( 2 k + 1 − x ) L k ( x ) − k L k − 1 ( x ) k + 1 . {\displaystyle L_{k+1}(x)={\frac {(2k+1-x)L_{k}(x)-kL_{k-1}(x)}{k+1}}.}
Furthermore,
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